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Question: problem 7 a recall that a product is associative if...

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Problem 7. (a) Recall that a product is associative if (abeab. Write down three generic matrices (a) (bij)ij-1, and (j)-, and check associativity of their product using the definition of matrix multiplication. (Conclude that it is much better to argue that matrix multiplication is just function (b) Draw a graph whose vertices are the five ways to parenthesize the product abcd, with an edge between two elements if they are connected by a single application of the associativity rule for three elements (for example, (ab)c)d has an edge to (ab) (cd)), since we can treat (ab) as one element, c as another, and d as the third). Do the same for the fourteen ways to parenthesize abcde. Conclude that associativity for a product of three elements implies associativity for four elements and five elements (c) Prove that associativity for a product of three elements implies associativity for any number of elements. What does this result mean in terms of the graphs from part (b)?

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